Finite Group Databasebeta

← #44560760

Subgroups — #44560760

Order 32128.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632238C211yesmaximal, characteristic
2C4⋊Q8×D251C211yesmaximal, characteristic, complemented
3#30632048C214yesmaximal, complemented
4#30632037C211yesmaximal, characteristic, complemented
5#30632035C211yesmaximal, characteristic, complemented
6#30632034C211yesmaximal, characteristic, complemented
7#30631687C214yesmaximal, complemented
8C4×#16860793C211yesmaximal, characteristic, complemented
9#487×C251C211yesmaximal, characteristic, complemented
10C2×#16860811C2212yes—
11C2×Q8×D251—21no—
12C2×#16860793C2212yes—
13#22548715C2211yescharacteristic
14#22548714—21no—
15#22548714C2212yes—
16#22548713—21no—
17#22548712C2211yescharacteristic
18#22548704C2212yes—
19#22548690—21no—
20#22548690—21no—
21#22548677—21no—
22#22548677—21no—
23C4⋊C4×D251—21no—
24C4⋊C4×D251C2212yes—
25C42×D251C2211yescharacteristic
26#22548639C2214yes—
27#22548637C2214yes—
28#22548635C2214yes—
29#22548625C2212yes—
30C4×C251⋊C8C2211yescharacteristic
31C4⋊Q8×C251C2211yescharacteristic
32C4⋊Q8×C251C2211yescharacteristic
33#103×C251C2214yes—
34C4×C8×C251C2211yescharacteristic
35#16860811—28no—
36C2×Dic502—22no—
37C2×Dic502—21no—
38Q8×D251—44no—
39C2×D1004—22no—
40C2×D1004—21no—
41C2×D1004—21no—
42C2×D1004C2312yes—
43C2×C4×D251—22no—
44C2×C4×D251—22no—
45C2×C4×D251D411yescharacteristic
46C2×C4×D251D411yescharacteristic
47C2×C4×D251C2311yescharacteristic
48#16860793—24no—
49C2×C251⋊C8C2312yes—
50C4⋊Dic251—21no—
51C4⋊Dic251C2312yes—
52Dic251⋊C4—22no—
53Dic251⋊C4—22no—
54(C2×D251)⋊C4—22no—
55(C2×D251)⋊C4—22no—
56C4×Dic251D411yescharacteristic
57C4×Dic251D411yescharacteristic
58C4×Dic251C2311yescharacteristic
59C2×Q8×C251—21no—
60C2×Q8×C251C2312yes—
61C2×C8×C251C2312yes—
62C4⋊C4×C251—21no—
63C4⋊C4×C251—21no—
64C4⋊C4×C251—21no—
65C4⋊C4×C251C2312yes—
66C42×C251C2311yescharacteristic
67C22×D251—22no—
68C22×D251C2×D411yescharacteristic
69Dic502—44no—
70Dic502—42no—
71D1004—24no—
72D1004—44no—
73D1004—42no—
74D1004—42no—
75C2×Dic251—22no—
76C2×Dic251C2×D411yescharacteristic
77C2×Dic251C2×D411yescharacteristic
78C2×Dic251C2×D411yescharacteristic
79C4×D251—22no—
80C4×D251—22no—
81C4×D251—44no—
82C4×D251—44no—
83C4×D251#4114yes—
84C251⋊C8—22no—
85Q8×C251—24no—
86Q8×C251—42no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2411yescharacteristic
92C8×C251—22no—
93C22×C251C22×D411yescharacteristic
94C2×D251—21no—
95C2×D251—21no—
96C2×D251—42no—
97C2×D251—42no—
98C2×D251—42no—
99C2×D251#12412yes—
100C4×C251—42no—
101C4×C251—42no—
102C4×C251AGL(1,ℤ/8)12yes—
103C4×C251#13712yes—
104C4×C251C2×#4112yes—
105Dic251—21no—
106Dic251—21no—
107Dic251—42no—
108Dic251#12412yes—
109C2×C251C2×#12411yescharacteristic
110C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C251C2×#13711yescharacteristic
112D251—22no—
113D251—82no—
114C251#314111yescharacteristic, complemented, Sylow
115#3141—2511nomaximal, Sylow
116#532—2511no—
117C2×C4⋊Q8—2511no—
118#489—2514no—
119#487—2511no—
120#487D25111yescharacteristic, complemented
121#486—2511no—
122#486—2511no—
123#417—2514no—
124C4×M16—2511no—
125C2×#41—2512no—
126C22×Q8—5021no—
127C2×M16—2512no—
128C4⋊Q8—5021no—
129C4⋊Q8—5021no—
130C4⋊Q8—2512no—
131C4⋊Q8—2511no—
132C4⋊Q8C2×D25111yescharacteristic
133C4⋊Q8C2×D25111yescharacteristic
134#127—2511no—
135#124—2512no—
136#121—5021no—
137#121—5021no—
138C4×D4—5021no—
139C4×D4—5021no—
140C2×C4⋊C4—5021no—
141C2×C4⋊C4—2512no—
142C2×C42—2511no—
143#103—2514no—
144#103C2×D25114yes—
145#102—2514no—
146#102—2514no—
147#97—2512no—
148C4×C8—2511no—
149C4×C8C2×D25111yescharacteristic
150#41—5028no—
151C2×Q8—21no—
152C2×Q8—10044no—
153C2×Q8—5022no—
154C2×Q8—5021no—
155C2×Q8C22×D25112yes—
156C2×D4—5022no—
157C2×D4—5021no—
158C2×D4—5021no—
159C2×D4—2512no—
160C22×C4—5022no—
161C22×C4—5022no—
162C22×C4—2511no—
163C22×C4—2511no—
164C22×C4—2511no—
165M16—5024no—
166C2×C8—2512no—
167C2×C8C22×D25112yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—5022no—
172C4⋊C4—5022no—
173C4⋊C4—5021no—
174C4⋊C4—2512no—
175C4⋊C4C22×D25112yes—
176C22⋊C4—5022no—
177C22⋊C4—5022no—
178C42—2511no—
179C42—2511no—
180C42—2511no—
181C42C22×D25111yescharacteristic
182C23—5022no—
183C23—2511no—
184Q8—24no—
185Q8—42no—
186Q8—10044no—
187Q8—10042no—
188D4—10044no—
189D4—10042no—
190D4—10042no—
191D4—5024no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—10044no—
195C2×C4—10044no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—2511no—
200C2×C4—2511no—
201C2×C4—2511no—
202C2×C4—2514no—
203C2×C4D4×D25111yescharacteristic
204C2×C4D4×D25111yescharacteristic
205C2×C4C23×D25111yescharacteristic
206C8—22no—
207C8—5022no—
208C22—10042no—
209C22—10042no—
210C22—10042no—
211C22—5021no—
212C22—5021no—
213C22—2512no—
214C22C2×D4×D25111yescharacteristic
215C4—42no—
216C4—42no—
217C4—10042no—
218C4—5021no—
219C4—5021no—
220C4—2512no—
221C4#2254874212yes—
222C4#2254874712yes—
223C4#41×D25112yes—
224C2—20082no—
225C2—5022no—
226C2#124×D25111yescharacteristic
227C2C2×#2254874211yescharacteristic
228C2C2×#2254874711yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#487×C251C2
socleC22×C251C22×D4
prime radical#30632238C2
hypercenter#487D251
hypocenterC251#3141
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C4⋊Q8×D251C2semidirect
3. #30632048C2semidirect
4. #30632037C2semidirect
5. #30632035C2semidirect
6. #30632034C2semidirect
7. #30631687C2semidirect
8. C4×#16860793C2semidirect
9. #487×C251C2semidirect
114. C251#3141semidirect
120. #487D251semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#306322382
C4⋊Q8×D2512
#306320482
#306320372
#306320352
#306320342
#306316872
C4×#168607932
#487×C2512
#3141251